All Questions
Tagged with tiling discrete-geometry
125 questions
10
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1
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151
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For what $n$ do there exist non-periodic tilings with rotational symmetry of order $n$?
More precisely, given an integer $n$, does there exist a non-periodic tiling, where there are infinitely many patches within the tiling, of indefinitely large area, with rotational symmetry of order $...
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20
votes
1
answer
2k
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Can you see through a cannonball packing?
More precisely, in a regular sphere packing, either the
HCP or FCC lattice packing,
does there exist a line $L$ disjoint from every sphere,
i.e., not touching any sphere?
If so, one could "look ...
- 151k
1
vote
0
answers
56
views
Tiling with one of each 3D shape
Encouraged by the positive solutions to my question,
Tiling with one of each shape,
I'd like to pose the $\mathbb{R}^3$ equivalent:
Q. Is there a tiling of $\mathbb{R}^3$ by (bounded) polyhedra, one ...
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2
votes
1
answer
235
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Tiling with one of each shape
Q. Is there a tiling of the plane by one each of simple polygons of $n$ vertices:
one triangle, one quadrilateral, one pentagon, $\ldots$ ,
one simple polygon of $n$ vertices, $\ldots$ ?
Here a ...
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0
votes
1
answer
98
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Chromatic tiling complexity and the chromatic number conjecture
Let $T$ be a finite set of tiles in $\mathbb{R}^d$. A tiling of $\mathbb{R}^d$ by $T$ is a collection of disjoint translates of tiles in $T$ whose union is $\mathbb{R}^d$. A tiling is $k$-chromatic if ...
2
votes
0
answers
62
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On convex polygons that can be cut into convex and mutually congruent pieces in exactly one way
Observations: any thin isosceles triangle has exactly 1 partition into 2 congruent pieces - only 1 line, bisector of its apex, does it.
By attaching a right triangle with base 1 and altitude 2 to an ...
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2
votes
2
answers
226
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On cutting tetrahedrons into mutually congruent pieces
Simple observations: A regular tetrahedron can be cut into 2 mutually congruent pieces (in 3 obvious ways which are all basically the same way, giving one and same pair of congruent pieces). The ...
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0
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0
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78
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Are there triangles that can be cut into 7 mutually congruent connected polygons?
First question below had appeared in a note at Triangles that can be cut into mutually congruent and non-convex polygons
Following the results of Beeson quoted in the answer at Subdivision of ...
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9
votes
1
answer
542
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Tracking a reference: "Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals"
I linked a paper by James Schmerl in a recent question which cites Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals, Privately Published, 1987.
I have had difficulty finding any ...
15
votes
1
answer
528
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Dividing a polyhedron into two similar copies
The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
1
vote
0
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52
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'Self-similar and perfect' partitions of planar regions
Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.
A classical example ...
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4
votes
1
answer
438
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Perfect squaring of rectangles
A perfect squaring of a rectangle may be defined as a partition of the rectangle into finitely many squares all of which are mutually non-congruent. https://en.wikipedia.org/wiki/Squaring_the_square ...
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0
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0
answers
41
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Trying to extend a theorem on Tiling with mutually non-congruent triangles
In the light of Cubing the cube - as 'perfectly' as possible, We try to slightly 'relax' the main theorem proved by Kupaavski, Pach and Tardos here:
https://arxiv.org/pdf/1711.04504.pdf
...
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9
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0
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186
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Cubing the cube - as 'perfectly' as possible
Ref: https://en.wikipedia.org/wiki/Squaring_the_square
A perfect cubing of a cube is a partition of the cube into some finite number of smaller cubes that are pair-wise non-congruent. The above page ...
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4
votes
0
answers
138
views
Hyponontiling Wang tiles
Call a finite collection of tiles that can tile the plane if we have to use each tile at least once tiling.
Is there a collection of at least 3 tiles that is not tiling, but such that after removing ...
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1
vote
1
answer
98
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To place copies of a planar convex region such that number of 'contacts' among them is maximized
A contact between two planar convex regions obviously happens either along a line segment or at a single point.
Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...
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1
vote
0
answers
92
views
Can a square be cut into non-right triangles that are mutually similar and pair-wise noncongruent?
We add a bit to Tiling the plane with pair-wise non-congruent and mutually similar triangles and Cutting polygons into mutually similar and non-congruent pieces
A (non square) rectangle can obviously ...
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4
votes
0
answers
175
views
Can a square be partitioned into mutually non-congruent triangles all of same area and perimeter?
It is known that the plane cannot be tiled by pair-wise non-congruent triangles all having same area and same perimeter (https://arxiv.org/abs/1711.04504).
Question: Can a square be partitioned into ...
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1
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0
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78
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To tile the plane with mutually non-congruent rational triangles of equal area
We add a little to Tiling the plane with pairwise non-congruent rational triangles
Question: Can the plane be tiled by pair-wise non-congruent rational triangles all of which have same area? If "...
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2
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0
answers
73
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Is this an actual solution for centroidal Voronoi tiling, or just a visual approximation? [closed]
For the capstone project at the end of my graduate Data Science studies in late 2022, I needed an algorithm that would converge to something close to centroidal Voronoi tiling, while tiling contiguous ...
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1
vote
0
answers
52
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Does a substitution tiling being FLC depend on starting seed?
I've been trying to understand more on "geometric" substitutions rather than just symbolic ones. As symbolic substitutions always yield FLC tilings, I wanted to know whether a tiling coming ...
- 633
1
vote
1
answer
138
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Recognizability/unique composition property for substitution tiling
This may be a very basic question, but I have not found an answer to it so far in my search. The question is whether there is an "algorithmic" way to check unique-composition/recognizability ...
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12
votes
0
answers
168
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Can the optimal packing density in $\mathbb{Z}^d$ be irrational?
For a finite $S \subset \mathbb{Z}^d$, let $d_p(S)$ be its optimal packing density. That is, the maximal lower asymptotic density of $A+S$, where $A \subset \mathbb{Z}^d$ is such that $(a_1+S)\cap (...
- 1,209
7
votes
1
answer
248
views
Decidability of completing Penrose tilings
Is the following problem known to be un/decidable? Problem: Given a finite configuration of Penrose tiles in the plane, determine if there is an extension of the configuration tiling the whole plane.
- 509
0
votes
1
answer
98
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Tiling the plane with quadrilaterals that are mutually non-congruent and affine equivalent
Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent?
Remark: Every trapezoid is affine equivalent to ...
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1
vote
0
answers
97
views
Tiling the plane with pair-wise non-congruent and mutually similar triangles
Question: Is it possible to tile the plane with triangles that are (1) mutually similar, (2) pairwise non-congruent and (3)non-right? No other constraints.
Note 1: Reg requirement 3 above: since any ...
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9
votes
4
answers
1k
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Tiling the plane with pairwise non-congruent rational triangles
A rational triangle is one in which all side lengths are rational numbers.
Question: Can we tile the Euclidean plane with rational triangles that are pairwise non-congruent? No further requirements on ...
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6
votes
1
answer
435
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On the aperiodic monotile
One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...
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17
votes
1
answer
580
views
Aperiodic monotile in $\mathbb{R}$
Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower ...
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1
vote
0
answers
37
views
Tiling the hyperbolic plane with mutually-non congruent equal area triangles
This post continues On tiling the plane with non-congruent, equal area triangles with each edge having a unique length
Can the hyperbolic plane be tiled by pair-wise non-congruent equal area ...
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1
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0
answers
46
views
Kissing behavior of planar regions
This post reworks a question that was stated in a slightly different form at Convex region $C$ with least kissing number of copies of $C$.
Background: Given a 2D region $C$ (not necessarily convex), ...
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8
votes
1
answer
248
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For which $n$ does a y-formed $n$-polyomino tile a $n \times n \times n$-cube?
I got from my children as a gift a puzzle consisting of 25 y-shaped 5-polyominoes that form a $5 \times 5 \times 5$-cube (see picture).
I'm wondering for which $n$ does a y-formed $n$-polyomino tile a ...
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5
votes
1
answer
397
views
How much of an aperiodic tiling is needed to force aperiodicity?
Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...
- 20.1k
15
votes
2
answers
778
views
How to characterize the regularity of a polygon?
In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
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7
votes
0
answers
87
views
Tiling the plane with mutually non-congruent equal area rectangles
Question: Is it possible to tile the plane with mutually non-congruent rectangles all of equal area?
Note 1: If the answer is "yes" then, there could be constrained versions of the question ...
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6
votes
2
answers
215
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Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...
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3
votes
1
answer
230
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Tiling planar integer lattice by finite point sets
I am interested in the following question.
Are there nice characterizations of the finite sets $S\subseteq \mathbb{Z}\times\mathbb{Z}$ that tile $ \mathbb{Z}\times\mathbb{Z}$ by translations (i.e. $\...
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2
votes
0
answers
208
views
Aperiodic tilings of the plane by squares and rhombi
Consider tilings of the plane by unit squares and by rhombi of unit side length and angles $\pi/3$, $2\pi/3$. It is easy to come up with periodic tilings of the plane - consider the following:
(from ...
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2
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0
answers
171
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Square-and-equilateral-triangle aperiodic tiling with $\leq 4$ prototiles?
There exist aperiodic tilings composed of square and equilateral-triangle tiles of unit side length: see https://tilings.math.uni-bielefeld.de/substitution/square-triangle/ and https://hal.archives-...
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1
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0
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62
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What are some other methods for partitioning an n-dimensional space based on a set of points in that space?
So this is a very general question, but I'm curious if there are any other methods for partitioning an n-dimensional space based on the location of a set of points, either randomly chosen or specified,...
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25
votes
1
answer
2k
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Polyomino that can tile itself
Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ ...
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2
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0
answers
78
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Tiling with triangles with same Steiner ellipses
We continue from Tiling with triangles of same circumradius and inradius .
Definitions: Given any triangle, its Steiner circumellipse is the unique circumellipse (ellipse that touches the triangle at ...
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42
votes
2
answers
2k
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Does any set of dominoes tile some common figure?
Let $D_1,\dots,D_n \subset \mathbb{Z}^2$ be two-point sets, i.e. 'dominoes' (unlike common dominoes, these are not necessarily connected, but I couldn't come up with a better name).
Does there always ...
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3
votes
1
answer
152
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Triangles that can be cut into mutually congruent and non-convex polygons
It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...
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1
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0
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40
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Tiling with a one-parameter family of non-congruent triangles
This post continues Tiling with triangles of same circumradius and inradius.
The following are known about infinite sets of triangles that can be parametrized with one variable:
from an infinite set ...
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5
votes
0
answers
177
views
Tiling with triangles of same circumradius and inradius
Consider a pair of positive real numbers $r$ and $R$ with $r<R/2$. Then we can form infinitely many triangles all with circumradius $R$ and inradius $r$.
For any such pair, the resulting triangles ...
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24
votes
3
answers
3k
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Polyomino that can cover an arbitrarily large square but not the entire plane
https://userpages.monmouth.com/~colonel/nrectcover/index.html
For a polyomino with no holes that cannot tile the plane, we may ask what are the maximal rectangles and infinite strips that it can ...
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7
votes
0
answers
227
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Tiling space with supertile of hypercube unfoldings
Two students in my class
asked and answered what might be a novel question.
It is well known that the cube has exactly $11$ edge-unfoldings
(or "nets"), as shown below:
(Image from ...
- 151k
2
votes
1
answer
145
views
Automorphism group of a normal tiling of the plane
A normal tiling of the plane is a tessellation for which every tile is topologically equivalent to a disk, the intersection of any two tiles is a connected set or the empty set, and all tiles are ...
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4
votes
2
answers
432
views
Family of shapes that can be tiled into one another
Okay, I'm trying to ask a question which hasn't been asked before, it may be futile, but let's see.
So let's take a square, this will be our shape A. We can tile a 2x1 rectangle by using shapes ...
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